In dry climates, in addition to air conditioners, people use "swamp coolers" or evaporative coolers that lower air temperature by evaporating water. They are used inland in southern california, too.
When water evaporates, it transforms heat energy into motion as water molecules break away and become a gas. A swamp cooler spreads water over a surface, usually a piece of fabric or shredded wood or coconut shells, and uses the cool surface to cool coils that, in turn, cool the interior air.
You can make a simple swamp cooler by putting your wet wash onto a drying rack, and then using a fan to blow air through the rack. The clothes will immediately cool down to below 70F, and in a few minutes, the air coming out the system will be cool.
Your clothes will also dry in a couple hours.
The only problem is that the water vapor will increase the humidity in the room, making you feel hotter. To fix the humidity problem, open the windows and maybe turn on the bathroom fan so the damp air is exchanged with dry air from outside. You'll need to be strategic to avoid bringing in very hot air.
If possible, orient everything so external air flows are passing through the clothes, onto you, and you may not even need to use the fan.
Dry air cools faster than humid air.
If you want to use this in conjunction with air conditioning, start off by using the swamp cooler to dry your clothes and bring the temperature down. Then, when the clothes are almost dry, close the windows and turn on the AC. The AC won't have to work as hard - and will also help dry out the air by condensing the water in the humid air (and then it will drip that over the coils to cool them).